Conic and projection checks
A fictional geometry worksheet defines an ellipse with semimajor axis 40 mm and center-to-focus distance 24 mm, using the standard horizontal major-axis Cartesian model. It also asks students to compare a central-perspective square behind a picture plane with a parallel-projection drawing. For the central example the square side is 40 mm, the viewing distance is 120 mm, and the square lies 40 mm behind the picture plane with its plane parallel to the picture plane. No general oblique measuring-point construction or arbitrary isometric angle-preservation claim is supplied. Students must show the ellipse arithmetic and state which perspective formula applies to the specified configuration. One draft calls every focal distance 80 mm and another says all parallel-projection angles retain their original values. The exercise concerns these explicit mathematical models rather than an official drawing-paper specification.
1. Calculate the ellipse semiminor axis and the apparent central-perspective side.
Read the worked answer
For the ellipse,b=sqrt(40²-24²)=32 mm, and the sum of distances from an ellipse point to the foci is 80 mm, not each distance. The central-perspective ratio is 120/(120+40)=0.75, giving a 30 mm apparent side in the stated model. Parallel projection preserves appropriate parallelism, but does not generally preserve angles; axis scale and projection orientation must be specified.
2. Correct the two overgeneralized statements in the draft.
Read the worked answer
For the ellipse,b=sqrt(40²-24²)=32 mm, and the sum of distances from an ellipse point to the foci is 80 mm, not each distance. The central-perspective ratio is 120/(120+40)=0.75, giving a 30 mm apparent side in the stated model. Parallel projection preserves appropriate parallelism, but does not generally preserve angles; axis scale and projection orientation must be specified.